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Null distribution
Null distribution
Null distribution
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In statistical hypothesis testing, the null distribution is the probability distribution of the test statistic when the null hypothesis is true.[1] For example, in an F-test, the null distribution is an F-distribution.[2] Null distribution is a tool scientists often use when conducting experiments. The null distribution is the distribution of two sets of data under a null hypothesis. If the results of the two sets of data are not outside the parameters of the expected results, then the null hypothesis is said to be true.

Null and alternative distribution

Examples of application

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The null hypothesis is often a part of an experiment. The null hypothesis tries to show that among two sets of data, there is no statistical difference between the results of doing one thing as opposed to doing a different thing. For an example of this, a scientist might be trying to prove that people who walk two miles a day have healthier hearts than people who walk less than two miles a day. The scientist would use the null hypothesis to test the health of the hearts of people who walked two miles a day against the health of the hearts of the people who walked less than two miles a day. If there was no difference between their heart rate, then the scientist would be able to say that the test statistics would follow the null distribution. Then the scientists could determine that if there was significant difference that means the test follows the alternative distribution.

Obtaining the null distribution

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In the procedure of hypothesis testing, one needs to form the joint distribution of test statistics to conduct the test and control type I errors. However, the true distribution is often unknown and a proper null distribution ought to be used to represent the data. For example, one sample and two samples tests of means can use t statistics which have Gaussian null distribution, while F statistics, testing k groups of population means, which have Gaussian quadratic form the null distribution.[3] The null distribution is defined as the asymptotic distributions of null quantile-transformed test statistics, based on marginal null distribution.[4] During practice, the test statistics of the null distribution is often unknown, since it relies on the unknown data generating distribution. Resampling procedures, such as non-parametric or model-based bootstrap, can provide consistent estimators for the null distributions. Improper choice of the null distribution poses significant influence on type I error and power properties in the testing process. Another approach to obtain the test statistics null distribution is to use the data of generating null distribution estimation.

Null distribution with large sample size

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The null distribution plays a crucial role in large scale testing. Large sample size allows us to implement a more realistic empirical null distribution. One can generate the empirical null using an MLE fitting algorithm.[5] Under a Bayesian framework, the large-scale studies allow the null distribution to be put into a probabilistic context with its non-null counterparts. When sample size n is large, like over 10,000, the empirical nulls utilize a study's own data to estimate an appropriate null distribution. The important assumption is that due to the large proportion of null cases ( > 0.9), the data can show the null distribution itself. The theoretical null may fail in some cases, which is not completely wrong but needs adjustment accordingly. In the large-scale data sets, it is easy to find the deviations of data from the ideal mathematical framework, e.g., independent and identically distributed (i.i.d.) samples. In addition, the correlation across sampling units and unobserved covariates may lead to wrong theoretical null distribution.[6] Permutation methods are frequently used in multiple testing to obtain an empirical null distribution generated from data. Empirical null methods were introduced with the central matching algorithm in Efron's paper.[7]

Several points should be considered using permutation method. Permutation methods are not suitable for correlated sampling units, since the sampling process of permutation implies independence and requires i.i.d. assumptions. Furthermore, literature showed that the permutation distribution converges to N(0,1) quickly as n becomes large. In some cases, permutation techniques and empirical methods can be combined by using permutation null replace N(0,1) in the empirical algorithm.[8]

References

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Null distribution

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In statistical hypothesis testing, the null distribution is the probability distribution of a test statistic under the assumption that the is true. It provides the theoretical foundation for evaluating how extreme an observed is, thereby enabling inferences about the validity of the . The null distribution is used to compute p-values, which represent the probability of obtaining a at least as extreme as the observed one assuming the holds, or to define rejection regions based on a chosen significance level (e.g., α = 0.05). For instance, in a one-sample t-test for a mean, the follows a t-distribution with n-1 under the null, allowing rejection of the null if the observed t-value falls in the tail beyond the critical value. Similarly, for tests of in contingency tables, the null distribution is often a , derived from the assumption of no association between variables. Null distributions can be derived analytically under parametric assumptions, such as normality via the for large samples, or approximated through methods like tests or when exact forms are unavailable. This flexibility ensures applicability across diverse statistical models, from simple means comparisons to complex regression analyses, while controlling the Type I error rate—the probability of falsely rejecting a true .

Fundamentals

Definition

In statistics, the null distribution refers to the of a under the assumption that the is true. It describes the expected sampling variability of the when there is no effect, no difference, or no association in the population, as specified by the H0H_0. The H0H_0 is a statement asserting the absence of an effect or relationship, such as equality of population means or independence of variables. A , which is a function derived from the sample to summarize against H0H_0, follows this null distribution when H0H_0 holds. In contrast, under the HAH_A, the would follow a different distribution, potentially shifting the probability mass to more extreme values. Mathematically, the null distribution is often expressed through the P(T≤t∣H0)P(T \leq t \mid H_0), where TT is the and tt is an observed value. The concept of the null distribution was introduced by R.A. Fisher in the and further developed within the Neyman-Pearson framework for testing, as outlined in their seminal 1933 paper on efficient tests of statistical hypotheses. This foundational work formalized the role of distributions under both null and alternative hypotheses to construct optimal decision rules. The null distribution underpins calculations by providing the baseline probability of extreme outcomes if H0H_0 is correct.

Role in Hypothesis Testing

In hypothesis testing, the null distribution serves as the foundational of the under the assumption that the H0H_0 is true, enabling researchers to quantify the evidence against H0H_0 based on observed data. It is used to compute the , defined as the probability of obtaining a at least as extreme as the observed value given H0H_0, such as P(T≥∣tobs∣∣H0)P(T \geq |t_{\text{obs}}| \mid H_0) for a two-sided test, where TT is the . This measures the compatibility of the data with the , with smaller values indicating stronger evidence against H0H_0. Additionally, the null distribution defines rejection regions, which are the tails of the distribution where the would lead to rejecting H0H_0 at a specified significance level α\alpha, such as the upper α\alpha- for a one-sided test. The null distribution directly controls the Type I error rate, or the probability of falsely rejecting H0H_0 when it is true, denoted as α=P(reject H0∣H0 true)\alpha = P(\text{reject } H_0 \mid H_0 \text{ true}). By setting α\alpha (commonly 0.05 or 0.01), the rejection region is calibrated so that the probability of a Type I error does not exceed this level under the null distribution, ensuring a controlled risk of false positives in the testing procedure. This framework, formalized in the Neyman-Pearson approach, emphasizes error rates over direct probability statements about hypotheses. The decision rule in hypothesis testing involves comparing the observed to critical values derived from the null distribution's ; for instance, in a right-tailed test, reject H0H_0 if the observed statistic exceeds the 1−α1 - \alpha of the null distribution. Equivalently, rejection occurs if the is less than or equal to α\alpha. This rule provides a systematic way to make inferences, balancing the risks of errors while relying on the null distribution for . The validity of the null distribution in hypothesis testing depends on key assumptions, including random sampling from the and the specified probabilistic model holding true under H0H_0, such as normality or of observations. Violations of these assumptions can distort the null distribution, leading to invalid p-values or error rates, underscoring the need for careful verification of preconditions before applying the testing framework.

Obtaining the Null Distribution

Analytical Derivation

Analytical derivation of the null distribution in parametric hypothesis testing follows a systematic process: first, specify the underlying statistical model, such as assuming independent and identically distributed observations from a known parametric family; second, state the null hypothesis H0H_0 that imposes restrictions on the parameters; third, construct a test statistic as a function of the data that captures deviations from H0H_0; and fourth, transform the statistic to a pivotal form whose distribution under H0H_0 does not depend on nuisance parameters, often by standardization or ratio formation, leading to a known distribution—exact in cases like the t or F under normality, or asymptotic like the chi-squared for goodness-of-fit tests. This approach relies on exact distributional properties under model assumptions, such as normality, to obtain closed-form expressions for the null distribution. In the parametric case of testing a population mean, consider independent observations X1,…,Xn∼N(μ,σ2)X_1, \dots, X_n \sim N(\mu, \sigma^2) with σ2\sigma^2 unknown. Under H0:μ=μ0H_0: \mu = \mu_0, the sample mean Xˉ\bar{X} satisfies n(Xˉ−μ0)/σ∼N(0,1)\sqrt{n} (\bar{X} - \mu_0)/\sigma \sim N(0,1)
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